Mathematics of Choice: Or, How to Count Without Counting by I Niven

By I Niven

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An integer a is a quadratic residue modulo a prime p if there exists x ∈ Z such that x2 ≡ a (mod p). Otherwise, a is a quadratic nonresidue modulo p. 134. The Legendre symbol for an integer a and a prime p is defined by   1 if a is a quadratic residue mod p and p ∤ a; a 0 if p | a; =  p −1 otherwise. , a p a p = b p = a+p p ab p and a2 p = 1 if p ∤ a. The Legendre symbol is multiplicative, . 135 (Euler’s criterion). For each odd prime p and integer a not divisible p−1 by p, a 2 ≡ ap (mod p).

YUG) Let f (a, b, c) = |b − a| b + a 2 |b − a| b + a 2 + − + + + . |ab| ab c |ab| ab c 40 3 Problems Prove that f (a, b, c) = 4 max{1/a, 1/b, 1/c}. 47. (ROU) Find the number of lines dividing a given triangle into two parts of equal area which determine the segment of minimum possible length inside the triangle. Compute this minimum length in terms of the sides a, b, c of the triangle. 48. (USS) Find all positive numbers p for which the equation x2 + px + 3p = 0 has integral roots. 49. (USS) Two mirror walls are placed to form an angle of measure α .

Let AB be a chord of a circle k and C its midpoint. Let p and q be two different lines through C that, respectively, intersect k on one side of AB in P and Q and on the other in P′ and Q′ . Let E and F respectively be the intersections of PQ′ and P′ Q with AB. Then it follows that CE = CF. 83. The power of a point X with respect to a circle k(O, r) is defined by P(X ) = OX 2 − r2 . For an arbitrary line l through X that intersects k at A and B −→ −→ (A = B when l is a tangent), it follows that P(X) = X A · X B.

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